$(50$ pts $)$ An engine cylinder contains $7{10}^{-5}kg$ of fuel with a heat of combustion, $q_c,$ of

$45,000k\frac{J}{k}g.$ The volume $V_2$ at top dead center is $0\mathrm{.}15{10}^{-3}{m}^3,$ and the volume $V_{\text{I}}$ at bottom

dead center is $1\mathrm{.}50{10}^{-3}{m}^3.$ The air$-$fuel ratio is $16$:$1,$ and the mixture temperature $T_l$ at the

start of compression is $300$ K $.$ Modeling the compression and combustion as an ideal gas

$(=1\mathrm{.}4,c_v=0\mathrm{.}71\frac{kJ}{k}gK)$ Otto cycle,

a$.$ What is the maximum temperature $T_3$ and pressure $P_3?$

b$.$ What is the pressure $P_l$ at the start of compression?